Two Particle States in an Asymmetric Box and the Elastic Scattering Phases

Chuan Liu; Xin Li; Xu Feng

REVIEW

Two Particle States in an Asymmetric Box and the Elastic Scattering Phases

Xu Feng, Xin Li, Chuan Liu

High Energy Physics - Lattice · Submitted 1 Apr 2004 · 18 references

Pith's verdict: A useful extension of Lüscher's formula to rectangular boxes, with a real gap in the D2 appendix that leaves Eq. (18) under-specified.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 2 major objections · 7 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-lat/0404001 (version 1)

Pith ID3IKKRTBK, Pith's permanent record of this paper and its review

CitationChuan Liu, Xin Li, Xu Feng. "Two Particle States in an Asymmetric Box and the Elastic Scattering Phases." arXiv:hep-lat/0404001 (2004). https://pith.science/paper/3IKKRTBK

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Bottom line: the D4 half of this is solid and useful; the D2 half has a hole in the appendix that the authors should be made to fill before anyone tries to use it. What's actually new: Lüscher's determinant formula is cubic-box only. Feng, Li and Liu redo the symmetry reduction for a rectangular box. Table 1 gives the basis polynomials for D4 and D2, Appendix A generalizes the modified zeta function to anisotropic lattice sums, Appendix B tabulates the reduced matrix elements, and Section 4 gives the large-L expansion with the c1, c2 coefficients in Table 2. That table is the part people will actually use: it lets you pull a scattering length out of an energy shift in, say, a 2:1:1 box. The derivation follows Lüscher's template closely and I don't see anything wrong with the D4 result. The soft spot is real and it sits in the D2 case. The D2 A+ sector is the union of the D4 A1+ and B1+ sectors, so its reduced matrix has cross terms — m_{0¯2}, m_{2¯2}, m_{0¯4}, m_{2¯4}, m_{4¯4}, m_{¯2◦4}, m_{¯4◦4} — that vanish identically when η1 = η2 and therefore appear in neither Table B.1 nor B.2. They are not zero for η1 ≠ η2: m_{0¯2} is proportional to W_{22}, which is nonzero as soon as the box is anisotropic. Eq. (18) and Eq. (21) use these elements (m_{2¯2} is right there in the determinant), but the text says the D2 elements 'are in fact the same as listed in Table B.1 and Table B.2' with renamed irreps. That is not true for the cross terms. Everything is still computable from Eq. (8), so this is a specification gap rather than a wrong result, but as printed Eq. (18) cannot be evaluated from the paper alone. Smaller things: the abstract says 'exact' while the determinants are truncated at l ≤ 4; Eq. (20) contains an m_{20¯4}/(cotδ4 − m_{¯4¯4}) term although the D4 A1+ basis has |◦4⟩ and no |¯4⟩, which looks like a typo; and there is no numerical check against a solvable model, which is exactly the kind of check that would have caught the appendix gap. Who should read it: lattice people doing hadron-hadron scattering or K→ππ at non-zero momentum who want more low-lying modes at fixed L. It is a methods paper and that community will cite it. Recommendation: send it out. The D4 result stands; the D2 appendix needs completing before the paper is usable as advertised.

Referee report

Major objections (2)

[Section 3 and Appendix B, Eq. (18)] The D2 A+ sector is not a single D4 irreducible representation but the union of D4 A+1 and B+1 (and A+2 at l = 4), so Eq. (18) requires cross terms between these D4 irreps that vanish identically in D4 and therefore appear in neither Table B.1 nor Table B.2. Specifically, the entries m_{0¯2}, m_{2¯2}, m_{0¯4}, m_{2¯4}, m_{4¯4}, m_{¯2◦4} and m_{¯4◦4} are needed for the 6×6 determinant but are absent from the appendix. These are generally non-zero: for example, direct evaluation from Eq. (8) gives m_{0¯2} = −√2 W_{22}, which is non-zero whenever Z_{22} ≠ 0. The statement in Appendix B that the D2 matrix elements are 'in fact the same as listed in Table B.1 and Table B.2' with renamed irreps is therefore not correct for the A+ sector, and the claim in Section 3 that the explicit expressions 'can be found in appendix B' is not borne out. Since Eq. (18) is one of the two central results, the missing elements should be computed and listed, or the appendix should explicitly state that they follow from Eq. (8) and give their expressions.
[Section 4, Eqs. (20)–(22) and (26)] The claim that the d-wave and g-wave contributions on the right-hand side of Eqs. (20) and (21) are 'smaller by a factor of q^2 compared with m00' is not justified for asymmetric boxes. In the cubic case the mixing matrix element m20 vanishes at q = 0 because Z20(1, 0) = 0 by cubic symmetry, so m20 ∼ q^2 and the correction is suppressed. For η1 ≠ η2, and also for η1 = η2 ≠ 1, the lattice symmetry is only D4 or D2, Y20 is invariant under that symmetry, and Z20(1, 0) is generally non-zero; then m20 ∼ 1/q^3 and the correction m20^2/(cotδ2 − m22) is of the same order in q as m00 ∼ 1/q. The same concern applies to the g-wave mixing. This affects the accuracy of the leading-order relation Eq. (22) and hence the scattering-length formula Eq. (26): the geometric s-wave/d-wave mixing is not suppressed by the d-wave scattering phase and is not accounted for by the argument that tanδ2 ∼ a2 k^5. The authors should either prove that the mixing vanishes for the aspect ratios considered, or quantify the size of the correction and state the conditions under which Eq. (22) and Eq. (26) are reliable.

Minor objections (7)

[Section 1] The carry-over of the non-relativistic quantum-mechanical relation to relativistic massive field theory is inherited from Lüscher's cubic-box analysis (Ref. [3]) and is not re-derived here. The manuscript should state more explicitly which features of that carry-over depend on the cubic topology and which do not, and whether the condition R < L/2 and the neglect of exponential finite-volume corrections are modified for the asymmetric aspect ratios tabulated in Table 2.
[Section 4, Eq. (20)] In the D4 A+1 equation the third term is printed with a subscript '¯4' and a denominator 'cotδ4 − m¯4¯4', but the D4 A+1 basis in Table 1 is |0⟩, |2⟩, |4⟩, |◦4⟩; the state |¯4⟩ belongs to B+1 and does not appear in this sector. The intended term is presumably m_{0◦4}^2/(cotδ4 − m_{◦4◦4}). The notation should be corrected and checked against Table 1.
[Section 4, Eq. (26)] The sentence that 'functions c1 and c2 receive contributions that are proportional to (a2/L^5)' is confusing, because c1 and c2 are defined as coefficients in the large-L expansion and should not themselves depend on L. The intended statement is presumably that the correction to the formula is of order a2/L^5 relative to the leading term. This should be reworded.
[Table 2] The numerical values of the subtracted zeta functions and the coefficients c1 and c2 are quoted to six decimal places without any estimate of the numerical precision or a convergence check of the series in Appendix A. A brief statement of the numerical accuracy, and ideally a comparison with an independent evaluation for at least one aspect ratio, would strengthen confidence in the table.
[Figs. 1 and 2] The figures would be clearer with explicit axis labels and units, and with a sentence in the caption explaining what the 'almost coincidence of singularities' means physically and for which values of q^2 it occurs. The lower panels of Fig. 2 are described as magnified versions of the upper panels, but the vertical scales are not stated in the caption.
[General] The formalism is presented for two distinguishable particles. For applications to ππ scattering mentioned in the introduction, the two particles are identical and the two-particle state must be symmetrized, which restricts the allowed partial waves and may modify the determinant conditions. The manuscript should state whether the derivation applies directly to identical particles or whether an additional symmetrization step is required.
[Abstract and Section 1] The word 'exact' in the abstract and introduction should be qualified: the determinant conditions are exact only in the sense of an infinite partial-wave expansion, while Eqs. (16) and (18) truncate at l = 4 and the derivation neglects exponential finite-volume corrections. A short clarification would avoid overstating the result.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

The paper derives determinant conditions that convert two-particle energy levels in a periodic rectangular box into continuum elastic scattering phases, generalizing Lüscher's cubic-box formula to $D_4$ and $D_2$ symmetry.

The reading

Lattice QCD can measure the energy of two interacting particles in a finite periodic box, but what one wants is the scattering phase shift in infinite volume. Lüscher showed how to convert cubic-box energies into phase shifts; this paper asks what happens when the box is a rectangular box of side lengths $(η_1L)×(η_2L)×L$ with periodic boundaries in all three directions. The authors derive determinant conditions, one for the $D_4$ case $η_1=η_2≠1$ and one for the $D_2$ case with all three sides unequal, that relate the box energy to the phases $δ_ℓ$ for angular momenta up to $ℓ=4$. In the low-momentum limit the s-wave phase dominates and the s-wave scattering length is tied to the energy shift by a large-$L$ expansion whose coefficients are tabulated for several aspect ratios. The practical motivation is that an asymmetric box offers more distinct low-lying momentum modes for the same shortest side, which can make non-zero relative momentum scattering cheaper to reach.

What carries the argument

The load-bearing device is the set of singular periodic solutions of the Helmholtz equation, built from $G_{ℓm}=Y_{ℓm}(∇)G$ with a periodic Green's function, and packaged into the modified zeta function $Z_{ℓm}(s,q^2;η_1,η_2)$ summed over the box momentum lattice $ñ=(n_1/η_1,n_2/η_2,n_3)$. Group theory then reduces the infinite matrix to blocks labelled by $D_4$ or $D_2$ irreducible representations, and the phase shifts enter through $U=(M+i)/(M-i)$ in the condition $det[e^{2iδ}-U]=0$. This is what turns a measured finite-box energy into a statement about continuum scattering.

Extended reading notes

Core claim

The central claim is that the exact two-particle energy eigenstates in a generic asymmetric rectangular box are still governed by a Lüscher-type determinant. For $D_4$ symmetry the $A_1^+$ sector gives a $4×4$ determinant involving $δ_0$, $δ_2$, and $δ_4$; for $D_2$ symmetry the $A^+$ sector gives a $6×6$ determinant because the reduced symmetry admits more independent basis polynomials. At low relative momentum $q$ both reduce to $δ_0(q)=Z_{00}(1,q^2;η_1,η_2)/(π^{3/2}η_1η_2 q)$, so the s-wave scattering length follows from $δE=-2πa_0/(η_1η_2μL^3)[1+c_1(η_1,η_2)(a_0/L)+c_2(η_1,η_2)(a_0/L)^2+⋯]$. The paper tabulates $c_1$ and $c_2$ for several aspect ratios and recovers the cubic-box values at $η_1=η_2=1$.

Load-bearing premise

The derivation is carried out in non-relativistic quantum mechanics and then asserted to carry over to relativistic massive field theory under the assumptions that the interaction range satisfies $R<L/2$ and that exponential finite-volume corrections can be neglected; that carry-over is inherited from Lüscher's framework rather than re-derived for the asymmetric topology.

Editorial extensions

If this is right

For a $D_4$ box with $η_1=η_2≠1$, an $A_1^+$ two-particle energy level determines $δ_0$ through a $4×4$ determinant whose off-diagonal entries are the $d$- and $g$-wave admixtures.
For a $D_2$ box with all sides unequal, the $A^+$ sector gives a $6×6$ determinant; at low $q$ it collapses to the same s-wave relation, $δ_0=Z_{00}/(π^{3/2}η_1η_2 q)$.
The s-wave scattering length follows from the energy-shift formula with $c_1,c_2$ tabulated for aspect ratios such as $6:5:4$, $4:3:2$, and $2:1:1$; at $1:1:1$ the cubic-box values are recovered.
Other symmetry sectors give access to higher partial waves: $E^+$, $B_1^+$, and $B_2^+$ for the $d$-wave in $D_4$, and $A_2^+$ for the $g$-wave.
An asymmetric box provides more distinct low-lying momentum modes than a cubic box of the same shortest side, so non-zero relative momentum scattering may be reachable with smaller lattices.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The sector-by-sector decomposition suggests a single asymmetric lattice could supply several partial-wave phase-shift points at once, since $E^+$, $B_1^+$, and $B_2^+$ levels probe the $d$-wave; the paper does not carry out that multi-sector extraction.
The tabulated $c_1$ and $c_2$ vary substantially with aspect ratio, so a natural practical question is whether some aspect ratio minimizes finite-volume systematics for a fixed shortest side; the paper lists values but does not optimize.
Because the relativistic carry-over is inherited rather than re-derived, a natural check is to repeat the derivation in a relativistic effective field theory or compare against a known-phase model in an asymmetric box.
If exponential finite-volume corrections grow with anisotropy, the gain in low-momentum modes could be offset by harder-to-control systematics; quantifying that trade-off is left open.

Original abstract

The exact two-particle energy eigenstates in a generic asymmetric rectangular box with periodic boundary conditions in all three directions are studied. Their relation with the elastic scattering phases of the two particles in the continuum are obtained for both $D_4$ and $D_2$ symmetry. These results can be viewed as a generalization of the corresponding formulae in a cubic box obtained by L\"uscher before. In particular, the s-wave scattering length is related to the energy shift in the finite box. Possible applications of these formulae are also discussed.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,161 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.08 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The exact two-particle energy eigenstates in a generic asymmetric rectangular box with periodic boundary conditions in all three directions are studied. Their relation with the elastic scattering phases of the two particles in the continuum are obtained for both $D_4$ and $D_2$ symmetry. These results can be viewed as a generalization of the corresponding formulae in a cubic box obtained by L\"uscher before. In particular, the s-wave scattering length is related to the energy shift in the finite box. Possible applications of these formulae are also discussed.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The exact two-particle energy eigenstates in a generic asymmetric rectangular box with periodic boundary conditions in all three directions are studied.	contribution 1.00	0.90	0.07	asserted 0.99
s1	Their relation with the elastic scattering phases of the two particles in the continuum are obtained for both $D_4$ and $D_2$ symmetry.	result 0.85	0.89	0.50	asserted 1.00
s2	These results can be viewed as a generalization of the corresponding formulae in a cubic box obtained by L\"uscher before.	interpretation 0.76	0.37	0.20	asserted 0.57
s3	In particular, the s-wave scattering length is related to the energy shift in the finite box.	result 0.96	0.72	0.21	asserted 1.00
s4	Possible applications of these formulae are also discussed.	future work 0.80	0.54	0.04	hedged possibility 0.75

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The functions σ2(q 2 ) and σ4(q 2 ) as a function of q 2 are plotted for both D4 symmetry with parameters η1 = η2 = 2. 1 2 3 4 5 0 20 40 60 80 100 q 2 σ 2 1 2 3 4 5 0 50 100 150 200 q 2 σ 4 1 2 3 4 5 0 2 4 6 8 10 q 2 σ 2 1 2 3 4 5 0 5 10 15 q 2 σ 4 2.0−1.5(Entire) 2.0−1.5(Part)

**Figure 2.** The functions σ2(q 2 ) and σ4(q 2 ) as a function of q 2 are plotted for both D2 symmetry with parameters η1 = 2, η2 = 1.5. where the angle φ(q) is defined to via: − tan φ(q) = 1/m00(q). The functions σ2(q) and σ4(q) represents the sensitivity of higher scattering phases. For D4 and D2 symmetry, they are given by: 11

Figures reproduced from arXiv:hep-lat/0404001 by the authors.

References

Works cited (18)

[1]

M. L¨ uscher. Commun. Math. Phys. , 105:153, 1986

View on Pith 1986
[2]

M. L¨ uscher and U. Wolﬀ. Nucl. Phys. B , 339:222, 1990

View on Pith 1990
[3]

M. L¨ uscher. Nucl. Phys. B , 354:531, 1991

View on Pith 1991
[4]

M. L¨ uscher. Nucl. Phys. B , 364:237, 1991

View on Pith 1991
[5]

M. Goeckeler, H.A. Kastrup, J. Westphalen, and F. Zimmer mann. Nucl. Phys. B, 425:413, 1994

View on Pith 1994
[6]

R. Gupta, A. Patel, and S. Sharpe. Phys. Rev. D , 48:388, 1993

View on Pith 1993
[7]

M. Fukugita, Y. Kuramashi, H. Mino, M. Okawa, and A. Ukawa . Phys. Rev. D, 52:3003, 1995

View on Pith 1995
[8]

S. Aoki et al. Nucl. Phys. (Proc. Suppl.) B , 83:241, 2000

View on Pith 2000
[9]

JLQCD Collaboration. Phys. Rev. D , 66:077501, 2002

View on Pith 2002
[10]

C. Liu, J. Zhang, Y. Chen, and J.P. Ma. Nucl. Phys. B , 624:360, 2002

View on Pith 2002
[11]

K.J. Juge. hep-lat/0309075, 2003

View on Pith Pith review arXiv 2003
[12]

CP-PACS Collaboration. Phys. Rev. D , 67:014502, 2003

View on Pith 2003
[13]

N. Ishizuka and T. Yamazaki. hep-lat/0309168, 2003

View on Pith Pith review arXiv 2003
[14]

CP-PACS Collaboration. hep-lat/0309155, 2003

View on Pith Pith review arXiv 2003
[15]

N. Ishizuka. hep-lat/0209108, 2002

View on Pith Pith review arXiv 2002
[16]

M. Rotenberg, R. Bivins, N. Metropolis, and J.K. Wooten Jr. The 3-j and 6-j Symbols . The Technology Press, Massachusetts Institute of Technol ogy, Cambridge, Massachusetts, USA, 1959

View on Pith 1959
[17]

L.D. Landau and E.M. Lifshitz. Quantum Mechanics, 3rd ed. Pergamon Press, Oxford, UK

View on Pith
[18]

J. D. Jackson. Classical Electrodynamics. John Wiley & Sons Inc., New York, USA, 1975. 18

View on Pith 1975

Forward citations

Cited by 29 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Higher order quantization conditions for two-body scattering with spin
hep-lat 2026-02 accept novelty 7.0 of 10

Higher-order finite-volume quantization conditions for spin-1/2 + spin-0 scattering are derived to J=11/2 and numerically checked to agree with independent box spectra to six significant figures.
Coupled-Channel $D\pi$, $D\eta$ and $D_{s}\bar{K}$ Scattering from Lattice QCD
hep-lat 2016-07 accept novelty 7.0 of 10

Coupled-channel lattice QCD at a pion mass of 391 MeV yields a near-threshold J^P = 0+ bound state, a deeply bound 1- state, and a narrow 2+ resonance in isospin-1/2 D-pi, D-eta and D_s-Kbar scattering.
An $a_0$ resonance in strongly coupled $\pi \eta$, $K\overline{K}$ scattering from lattice QCD
hep-ph 2016-02 conditional novelty 7.0 of 10

Lattice QCD spectra in three volumes yield a coupled-channel pi-eta, K-Kbar S-matrix whose cusp at K-Kbar threshold is traced to a single resonance pole on one unphysical Riemann sheet.
Two-nucleon systems at $m_{\pi}\approx292$ MeV from lattice QCD
hep-lat 2026-05 conditional novelty 6.0 of 10

Lattice QCD at m_pi≈292 MeV finds virtual poles in the ^3S1 and ^1S0 NN channels with binding energies 6^{+5}_{-3} MeV and 11^{+6}_{-5} MeV, extracted via Lüscher and NPHF analyses.
A variational study of two-nucleon systems with lattice QCD
hep-lat 2021-08 conditional novelty 6.0 of 10

Variational lattice QCD with up to 42 two-nucleon operators yields upper bounds on dineutron and deuteron energy levels at m_pi = 806 MeV, and shows that omitting operator families can delete energy levels from the ex...

Show 24 more citing papers

Low-energy Scattering and Effective Interactions of Two Baryons at $m_{\pi}\sim 450$ MeV from Lattice Quantum Chromodynamics
hep-lat 2020-09 conditional novelty 6.0 of 10

At a pion mass of about 450 MeV, lattice QCD shows most two-baryon channels bind, while spin-triplet Sigma-N and Xi-Xi repel, and the data support approximate SU(6) spin-flavor symmetry.
Scattering amplitudes from finite-volume spectral functions
hep-lat 2019-03 conditional novelty 6.0 of 10

A modified LSZ reduction applied to finite-volume spectral functions smeared with a complex iε kernel yields exclusive m-to-n scattering amplitudes in the ordered limit of large volume followed by vanishing kernel width.
Dynamically-coupled partial-waves in $\rho\pi$ isospin-2 scattering from lattice QCD
hep-lat 2018-02 accept novelty 6.0 of 10

Lattice QCD spectra of 141 energy levels determine the coupled 3S1 and 3D1 rho-pion scattering amplitudes and a small but non-zero S-D mixing angle in isospin-2.
Baryon-Baryon Interactions and Spin-Flavor Symmetry from Lattice Quantum Chromodynamics
hep-lat 2017-06 accept novelty 6.0 of 10

Lattice QCD at a pion mass of 806 MeV finds two-baryon scattering parameters consistent with the SU(6) spin-flavor symmetry predicted by large-Nc QCD, with a numerical hierarchy suggesting an accidental SU(16) symmetry.
Multichannel one-to-two transition amplitudes in a finite volume
hep-lat 2014-06 accept novelty 6.0 of 10

A generalized Lellouch-Luscher formula converts finite-volume lattice matrix elements of a current between a one-particle and a two-particle state into infinite-volume transition amplitudes, for any number of coupled ...
Two-particle multichannel systems in a finite volume with arbitrary spin
hep-lat 2014-01 accept novelty 6.0 of 10

A single master equation (Eq. 22) gives the finite-volume quantization condition for two-particle systems with arbitrary spin, arbitrary open coupled channels, arbitrary momentum, and twisted or periodic boundary conditions.
Two-Baryon Systems with Twisted Boundary Conditions
hep-lat 2013-11 conditional novelty 6.0 of 10

A quarter-turn twist in each spatial direction suppresses finite-volume corrections to the deuteron binding energy from order exp(-kappa L)/L to exp(-2 kappa L)/L, and cuts nucleon-mass corrections by a factor of three.
Two-Nucleon Systems in a Finite Volume: (I) Quantization Conditions
hep-lat 2013-05 accept novelty 6.0 of 10

The paper derives and tabulates finite-volume quantization conditions relating two-nucleon lattice energy eigenvalues to S-, P-, D-, and F-wave scattering parameters, including partial-wave mixing, for three boost classes.
n-Boson Energies at Finite Volume and Three-Boson Interactions
hep-lat 2007-07 conditional novelty 6.0 of 10

The volume dependence of the n-boson ground-state energy is extended to order 1/L^6, the order at which the two-body effective range and the three-body contact interaction first contribute.
Form factors of two-hadron states from a covariant finite-volume formalism
hep-lat 2018-12 conditional novelty 5.0 of 10

A Lorentz-covariant reformulation of the finite-volume 2+J to 2 formalism, with a new finite-volume function G whose numerical evaluation is worked out in detail, including its triangle singularities.
The $\pi\gamma \to \pi\pi$ transition and the $\rho$ radiative decay width from lattice QCD
hep-lat 2018-07 conditional novelty 5.0 of 10

Lattice QCD yields the pi-gamma to pi-pi transition amplitude across photon virtuality and dipion mass, and from its rho-pole residue the photocoupling |G_rho-pi-gamma| = 0.0802(32)(20), roughly 1.6 times the experime...
Extraction of isoscalar $\pi\pi$ phase-shifts from lattice QCD
hep-lat 2018-03 conditional novelty 5.0 of 10

Lattice QCD at two pion masses yields isoscalar pion-pion phase shifts, and a chiral unitary extrapolation puts the sigma pole at (440 - i 240) MeV at the physical point.
Rho resonance parameters from lattice QCD
hep-lat 2016-05 conditional novelty 5.0 of 10

Two-flavor lattice QCD phase shifts extrapolate to a rho mass of 720 MeV, roughly 50 MeV below experiment, with the shortfall attributed to the missing strange quark through a unitarized chiral model.
$I=1$ and $I=2$ $\pi-\pi$ scattering phase shifts from $N_{\mathrm{f}} = 2+1$ lattice QCD
hep-lat 2016-04 conditional novelty 5.0 of 10

On a single 3.7 fm lattice ensemble with 230 MeV pions, stochastic LapH quark propagation yields m_rho/m_pi = 3.350(24), g_rhoppi = 5.99(26), and a 20%-precision I=2 scattering length, at roughly 1/170 the Dirac inver...
Hadron-Hadron Interactions from $N_f=2+1+1$ Lattice QCD: isospin-2 $\pi\pi$ scattering length
hep-lat 2015-06 conditional novelty 5.0 of 10

The first Nf=2+1+1 lattice QCD determination gives Mπa0 = -0.0442(2)stat(+4-0)sys for the isospin-2 ππ scattering length, consistent with chiral perturbation theory and prior lattice results.
Two-Nucleon Systems in a Finite Volume: (II) 3S1-3D1 Coupled Channels and the Deuteron
hep-lat 2013-09 conditional novelty 5.0 of 10

Finite-volume spectra of boosted two-nucleon systems in 10-14 fm boxes are predicted to constrain the deuteron binding energy, scattering length, effective range, and S-D mixing angle from a single lattice QCD ensemble.
Lattice QCD study of the s-wave $\pi\pi $ scattering lengths in the I=0 and 2 channels
hep-lat 2013-03 conditional novelty 5.0 of 10

Lattice QCD with MILC staggered configurations yields mπ a^{I=0} = 0.214(4)(7) and mπ a^{I=2} = -0.04430(25)(40) at the physical pion mass, with the I=0 value including the disconnected diagram.
Resonance parameters of the rho-meson from asymmetrical lattices
hep-lat 2012-11 conditional novelty 5.0 of 10

Using three elongated lattice volumes at a pion mass of 304 MeV, the authors obtain m_rho = 827(3)(5) MeV and g_rho_pi_pi = 6.67(42) from six P-wave pion-pion scattering phase shifts.
Generalized L\"uscher Formula in Multi-channel Baryon-Meson Scattering
hep-lat 2012-09 accept novelty 5.0 of 10

A generalized Lüscher quantization condition is derived for two-channel baryon-meson scattering with a spin-1/2 particle, in center-of-mass and moving frames, and shown to agree between quantum mechanics and quantum f...
Towards grounding nuclear physics in QCD
nucl-th 2019-10 accept novelty 3.0 of 10

A review arguing that a quantitative bridge from QCD to nuclei requires lattice QCD, effective field theory, and ab initio many-body methods, with a candid account of the unresolved two-nucleon lattice controversy.
Lattice simulations for few- and many-body systems
nucl-th 2008-04 accept novelty 3.0 of 10

Lattice effective field theory provides a common computational framework for simulating cold atoms and low-energy nuclear systems, though sign problems and coarse lattice spacings still limit precision.
Multi-particle systems on the lattice and chiral extrapolations: a brief review
hep-lat 2021-02 accept novelty 2.0 of 10

Finite-volume quantization conditions combined with unitarized chiral perturbation theory form a workable pipeline from lattice energy levels to two- and three-hadron scattering amplitudes and resonance parameters, th...
Status and Progress of Hamiltonian Effective Field Theory in Light Hadron Spectroscopy
hep-ph 2026-10 conditional novelty 1.0 of 10

A survey of HEFT showing that lattice QCD spectra, rather than scattering data alone, discriminate between bare-core and dynamically generated interpretations of light hadron resonances.
Hadrons and Nuclei
hep-lat 2019-04 conditional novelty 1.0 of 10

A community roadmap arguing that lattice QCD has matured enough to connect quark and gluon dynamics quantitatively to hadron and nuclear observables, and laying out the calculations needed next.

How the literature uses it

43 arXiv papers cite this in 70 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 51 sentences without a confident label

Builds on it 2 papers

Elongated boxes have a different symmetry than cubic ones and different Lüscher formulas are required: for the zero-momentum case they were worked out in [ref] and in this paper we worked out the required one for states boosted along the elongated direction.

Rho resonance parameters from lattice QCD · Conclusions · 0.97
Then, this formula was generalized to moving frames with center-of-mass momentum $P = (2π/L) k$ [ref] , [equation] where [equation] is the generalized zeta function.

Galilean invariance restoration on the lattice · Lüscher's formula · 0.80

Uses its method or tool 4 papers

For energies below the inelastic threshold we use Lüscher's formula [ref] and the extensions to elongated boxes [ref] and boosted frames [ref] .

Pion scattering in the isospin I=2 channel from elongated lattices · Extracting physical quantities · 0.97
To connect the two-hadron state energies determined from lattice QCD with the physical observables, i.e., the phase-shifts in the continuum, we use Lüscher's formula [ref] and its extensions to the elongated box [ref] and to states boosted along the elongated direction [ref] .

Extraction of isoscalar $\pi\pi$ phase-shifts from lattice QCD · Phase-shift formulas · 0.98
Expanding Eq. ([ref]) by subsitituting Eqs. ([ref]) and ([ref]) follows: [equation] where $q= pL/2π$ is the dimensionless momentum, the term of $q δ$ can be determined from lattice calculations by Lüscher's method [ref] , and the zeta function $𝒵_00(s;q^2)$ is computed numerically [ref] .

First-Principles Determination of the Proton-Proton Fusion Matrix Element from Lattice QCD · Finite-Volume Analysis · 0.99

Compares with it 2 papers

It is evident from the result, that it is consistent with all previous two-body finite volume results [ref] .

Two-particle multichannel systems in a finite volume with arbitrary spin · Conclusion · 0.87
For the meson-meson scattering the formulas derived here for the states at rest agree with the one derived by in Ref. [ref] , while for the moving case they are mostly new.

Scattering phaseshift formulas for mesons and baryons in elongated boxes · Conclusion · 1.00

Cites it as background 8 papers

It has been generalized in various ways over the years: to the boosted frames [ref] , to the case of asymmetric boxes [ref] , with twisted boundary conditions [ref] , to the case of particles with spin, e.g. to the case of two baryons [ref] and beyond the inelastic threshold [ref] , and even to three-particle case [ref] .

Generalized L\"uscher's Formula in Multichannel Baryon-Baryon Scattering · INTRODUCTION · 0.92
The evaluation of these zeta functions is non-trivial but has been discussed in Refs. [ref] .

Higher order quantization conditions for two-body scattering with spin · Kinematics · 0.90
For systems with PBCs, there has been a great deal of effort in reduction of these master equation for a wide variety of scenarios [ref] .

Two-particle multichannel systems in a finite volume with arbitrary spin · Two-particle multichannel systems with spin and PBCs · 0.87

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

M. L¨ uscher. Commun. Math. Phys. , 105:153, 1986

View on Pith 1986

[2] [2]

M. L¨ uscher and U. Wolﬀ. Nucl. Phys. B , 339:222, 1990

View on Pith 1990

[3] [3]

M. L¨ uscher. Nucl. Phys. B , 354:531, 1991

View on Pith 1991

[4] [4]

M. L¨ uscher. Nucl. Phys. B , 364:237, 1991

View on Pith 1991

[5] [5]

M. Goeckeler, H.A. Kastrup, J. Westphalen, and F. Zimmer mann. Nucl. Phys. B, 425:413, 1994

View on Pith 1994

[6] [6]

R. Gupta, A. Patel, and S. Sharpe. Phys. Rev. D , 48:388, 1993

View on Pith 1993

[7] [7]

M. Fukugita, Y. Kuramashi, H. Mino, M. Okawa, and A. Ukawa . Phys. Rev. D, 52:3003, 1995

View on Pith 1995

[8] [8]

S. Aoki et al. Nucl. Phys. (Proc. Suppl.) B , 83:241, 2000

View on Pith 2000

[9] [9]

JLQCD Collaboration. Phys. Rev. D , 66:077501, 2002

View on Pith 2002

[10] [10]

C. Liu, J. Zhang, Y. Chen, and J.P. Ma. Nucl. Phys. B , 624:360, 2002

View on Pith 2002

[11] [11]

K.J. Juge. hep-lat/0309075, 2003

View on Pith Pith review arXiv 2003

[12] [12]

CP-PACS Collaboration. Phys. Rev. D , 67:014502, 2003

View on Pith 2003

[13] [13]

N. Ishizuka and T. Yamazaki. hep-lat/0309168, 2003

View on Pith Pith review arXiv 2003

[14] [14]

CP-PACS Collaboration. hep-lat/0309155, 2003

View on Pith Pith review arXiv 2003

[15] [15]

N. Ishizuka. hep-lat/0209108, 2002

View on Pith Pith review arXiv 2002

[16] [16]

M. Rotenberg, R. Bivins, N. Metropolis, and J.K. Wooten Jr. The 3-j and 6-j Symbols . The Technology Press, Massachusetts Institute of Technol ogy, Cambridge, Massachusetts, USA, 1959

View on Pith 1959

[17] [17]

L.D. Landau and E.M. Lifshitz. Quantum Mechanics, 3rd ed. Pergamon Press, Oxford, UK

View on Pith

[18] [18]

J. D. Jackson. Classical Electrodynamics. John Wiley & Sons Inc., New York, USA, 1975. 18

View on Pith 1975